I have attached the data. I want to fit the data with the model which is very complicated which contains double numerical integration and has parameters 'par'. I have the only choice of fitnlm as per my knowledge. But I think it might take much time to fit. If you could recommend other algorithm which do not require derivative during iteration such as differential evolutution or any other, pls tell me how to implement.
clc;
clear all;
load('Figure9S11.mat');
load('Figure9S22.mat');
xdata11 = Figure9S11(1:225,1);
ydata11 = Figure9S11(1:225,2);
beta0=[2;1;0.1;212.8];
% Fit the model to the data
mdl = fitnlm(xdata11, ydata11, @BiaxialNewInvariant11, beta0);
disp(mdl)
function K1 = BiaxialNewInvariant11(par,lambda1)
hmod33minushmod11 = @(theta,phi,lambda1) (1/2).*lambda1.^2.*cos(phi).^2.*sin(theta).^2.*(1+((-1)+ ...
lambda1.^2).*(lambda1.^(-8).*((-1)+lambda1.^2).^2.*((1+ ...
lambda1.^2).^2.*cos(theta).^2+lambda1.^8.*sin(theta).^2)).^( ...
-1/2))+(-1/2).*lambda1.^(-4).*cos(theta).^2.*(1+lambda1.^( ...
-4).*(1+(-1).*lambda1.^4).*(lambda1.^(-8).*((-1)+lambda1.^2) ...
.^2.*((1+lambda1.^2).^2.*cos(theta).^2+lambda1.^8.*sin( ...
theta).^2)).^(-1/2));
Imat = @(theta,lambda1) (1/4).*lambda1.^(-4).*(1+(-2).*lambda1.^4+lambda1.^6+(-1).*( ...
(-1)+lambda1.^6).*cos(2.*theta)+lambda1.^4.*(lambda1.^(-8).* ...
(2.*((-1)+lambda1.^2).^2.*(1+2.*lambda1.^2+lambda1.^4+ ...
lambda1.^8)+(-2).*((-1)+2.*lambda1.^4+(-2).*lambda1.^10+ ...
lambda1.^12).*cos(2.*theta))).^(1/2));
rho = @(theta,phi,par) 2.*2.^(1/2).*exp(1).^(2.*par(4).*cos(phi).^2.*sin(theta).^2).*( ...
par(4).*pi.^(-1)).^(1/2).*erfi(2.^(1/2).*par(4).^(1/2)).^(-1);
Dw = @(theta,lambda1,par) (3/2).*par(2).*exp(1).^(par(3).*Imat(theta,lambda1).^3).*Imat(theta,lambda1).^4;
integrand = @(par,lambda1,theta,phi) 2*par(1)*Dw(theta,lambda1,par).*(sin(theta)).*rho(theta,phi,par).*Dw(theta,lambda1,par).* ...
hmod33minushmod11(theta,phi,lambda1);
K1 = 0.3E0.*((-1).*lambda1.^(-4)+lambda1.^2)+ integral2(@(phi,theta) integrand(par,lambda1,theta,phi),0,2*pi,0,pi);
end
